X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week2.mdwn;h=b23d1a74efb87fabd5bda7a89ce16e9f3406a9d0;hp=a42ec6bf61fc028cd466a6b69da4ca2823faf88a;hb=d1761bf08a977c86230b82fbd4b9da0c7b940d78;hpb=1b911b52c1885ce930c310f13700188b09d9b840 diff --git a/week2.mdwn b/week2.mdwn index a42ec6bf..b23d1a74 100644 --- a/week2.mdwn +++ b/week2.mdwn @@ -36,11 +36,11 @@ Lambda expressions that have no free variables are known as **combinators**. Her > **K** is defined to be \x y. x. That is, it throws away its second argument. So K x is a constant function from any (further) argument to x. ("K" for "constant".) Compare K - to our definition of **true**. + to our definition of true. -> **get-first** was our function for extracting the first element of an ordered pair: \fst snd. fst. Compare this to **K** and **true** as well. +> **get-first** was our function for extracting the first element of an ordered pair: \fst snd. fst. Compare this to K and true as well. -> **get-second** was our function for extracting the second element of an ordered pair: \fst snd. snd. Compare this to our definition of **false**. +> **get-second** was our function for extracting the second element of an ordered pair: \fst snd. snd. Compare this to our definition of false. > **B** is defined to be: \f g x. f (g x). (So B f g is the composition \x. f (g x) of f and g.) @@ -48,7 +48,7 @@ Lambda expressions that have no free variables are known as **combinators**. Her > **W** is defined to be: \f x . f x x. (So W f accepts one argument and gives it to f twice. What is the meaning of W multiply?) -> **ω** is defined to be: \x. x x +> **ω** (that is, lower-case omega) is defined to be: \x. x x It's possible to build a logical system equally powerful as the lambda calculus (and readily intertranslatable with it) using just combinators, considered as atomic operations. Such a language doesn't have any variables in it: not just no free variables, but no variables at all. @@ -65,7 +65,7 @@ duplicators. ![reflexive](http://lambda.jimpryor.net/szabolcsi-reflexive.jpg) -Notice that the semantic value of *himself* is exactly W. +Notice that the semantic value of *himself* is exactly W. The reflexive pronoun in direct object position combines first with the transitive verb (through compositional magic we won't go into here). The result is an intransitive verb phrase that takes a subject argument, duplicates that argument, and feeds the two copies to the transitive verb meaning. Note that W <~~> S(CI): @@ -103,14 +103,14 @@ S takes three arguments, duplicates the third argument, and feeds one copy to th SFGX ~~> FX(GX) If the meaning of a function is nothing more than how it behaves with respect to its arguments, -these reduction rules capture the behavior of the combinators S,K, and I completely. -We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of S's and K's: +these reduction rules capture the behavior of the combinators S, K, and I completely. +We can use these rules to compute without resorting to beta reduction. For instance, we can show how the I combinator is equivalent to a certain crafty combination of Ss and Ks: SKKX ~~> KX(KX) ~~> X -So the combinator SKK is equivalent to the combinator I. +So the combinator SKK is equivalent to the combinator I. -Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S,K, and I as defined here. +Combinatory Logic is what you have when you choose a set of combinators and regulate their behavior with a set of reduction rules. The most common system uses S, K, and I as defined here. ###The equivalence of the untyped lambda calculus and combinatory logic### @@ -124,7 +124,7 @@ Assume that for any lambda term T, [T] is the equivalent combinatory logic term. 1. [a] a 2. [(M N)] ([M][N]) 3. [\a.a] I - 4. [\a.b] Kb assumption: a and b are different + 4. [\a.M] KM assumption: a does not occur free in M 5. [\a.(M N)] S[\a.M][\a.N] 6. [\a\b.M] [\a[\b.M]] @@ -134,12 +134,64 @@ The second rule says that the way to translate an application is to translate th first element and the second element separately. The third rule should be obvious. The fourth rule should also be fairly self-evident: since what a lambda term such as \x.y does it throw away its first argument and return y, that's exactly what the combinatory logic translation should do. And indeed, Ky is a function that throws away its argument and returns y. -The fifth rule breaks down an abstract whose body is an application. The S combinator takes its next argument and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambda in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.) +The fifth rule deals with an abstract whose body is an application: the S combinator takes its next argument (which will fill the role of the original variable a) and copies it, feeding one copy to the translation of \a.M, and the other copy to the translation of \a.N. Finally, the last rule says that if the body of an abstract is itself an abstract, translate the inner abstract first, and then do the outermost. (Since the translation of [\b.M] will not have any lambdas in it, we can be sure that we won't end up applying rule 6 again in an infinite loop.) [Fussy notes: if the original lambda term has free variables in it, so will the combinatory logic translation. Feel free to worry about this, though you should be confident that it makes sense. You should also convince yourself that if the original lambda term contains no free variables---i.e., is a combinator---then the translation will consist only of S, K, and I (plus parentheses). One other detail: this translation algorithm builds expressions that combine lambdas with combinators. For instance, the translation of \x.\y.y is [\x[\y.y]] = [\x.I] = KI. In that intermediate stage, we have \x.I. It's possible to avoid this, but it takes some careful thought. See, e.g., Barendregt 1984, page 156.] - -These systems are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation! +Here's an example of the translation: + + [\x\y.yx] = [\x[\y.yx]] = [\x.S[\y.y][\y.x]] = [\x.(SI)(Kx)] = S[\x.SI][\x.Kx] = S(K(SI))(S[\x.K][\x.x]) = S(K(SI))(S(KK)I) + +We can test this translation by seeing if it behaves like the original lambda term does. +The orginal lambda term lifts its first argument (think of it as reversing the order of its two arguments): + + S(K(SI))(S(KK)I) X Y = + (K(SI))X ((S(KK)I) X) Y = + SI ((KK)X (IX)) Y = + SI (KX) Y = + IY (KX)Y = + Y X + +Viola: the combinator takes any X and Y as arguments, and returns Y applied to X. + +Back to linguistic applications: one consequence of the equivalence between the lambda calculus and combinatory +logic is that anything that can be done by binding variables can just as well be done with combinators. +This has given rise to a style of semantic analysis called Variable Free Semantics (in addition to +Szabolcsi's papers, see, for instance, +Pauline Jacobson's 1999 *Linguistics and Philosophy* paper, Towards a variable-free Semantics'). +Somewhat ironically, reading strings of combinators is so difficult that most practitioners of variable-free semantics +express there meanings using the lambda-calculus rather than combinatory logic; perhaps they should call their +enterprise Free Variable Free Semantics. + +A philosophical application: Quine went through a phase in which he developed a variable free logic. + + Quine, Willard. 1960. Variables explained away. {\it Proceedings of + the American Philosophical Society}. Volume 104: 343--347. Also in + W.~V.~Quine. 1960. {\it Selected Logical Papers}. Random House: New + York. 227--235. + +The reason this was important to Quine is similar to the worries that Jim was talking about +in the first class in which using non-referring expressions such as Santa Clause might commit +one to believing in non-existant things. Quine's slogan was that to be is to be the value of a variable'. +What this was supposed to mean is that if and only if an object could serve as the value of some variable, we +are committed to recognizing the existence of that object in our ontology. +Obviously, if there ARE no variables, this slogan has to be rethought. + +Quine did not appear to appreciate that Shoenfinkel had already invented combinatory logic, though +he later wrote an introduction to Shoenfinkel's key paper reprinted in Jean +van Heijenoort (ed) 1967 *From Frege to Goedel, + a source book in mathematical logic, 1879--1931*. +Cresswell has also developed a variable-free approach of some philosophical and linguistic interest +in two books in the 1990's. + +A final linguistic application: Steedman's Combinatory Categorial Grammar, where the "Combinatory" is +from combinatory logic (see especially his 2000 book, *The Syntactic Process*). Steedman attempts to build +a syntax/semantics interface using a small number of combinators, including T = \xy.yx, B = \fxy.f(xy), +and our friend S. Steedman used Smullyan's fanciful bird +names for the combinators, Thrush, Bluebird, and Starling. + +Many of these combinatory logics, in particular, the SKI system, +are Turing complete. In other words: every computation we know how to describe can be represented in a logical system consisting of only a single primitive operation! Here's more to read about combinatorial logic. Surely the most entertaining exposition is Smullyan's [[!wikipedia To_Mock_a_Mockingbird]].